536 INMAN AND BAGNOLD [CHAP. 21 



net transport. The dependence of sediment transport on the degree of asym- 

 metry of orbital motion, when ripple size and orbital diameter are of similar 

 magnitude, suggests that the ratio of ripple wavelength to orbital displacement 

 is a measure of similitude that should be considered in models. 



The shoreward trans])()rt of sand effected by long stable waves outside the 

 ])lunge line is reduced or inhibited by an upward slope in the bed. This reduction 

 in transport appears to be readily explainable by the reasoning leading to 

 relation (18). A condition of tangential force equilibrium on the sand grains is 

 approached. Further, if this equilibrium condition were to be disturbed by a 

 reduction of the onshore water drift, due to a steepening of the waves, the result 

 should be a net offshore transport of sand down the slope. This would explain 

 the apparent anomaly; while oifshore-flowing bottom currents have never 

 been detected outside of the plunge line in two-dimensional models, offshore 

 sand transport has been reported both in models and in nature. 



In nature, high short waves also produce an offshore transport of sediment 

 by increasing the turbulence and velocity of the currents in the nearshore 

 circulation system, as discussed in a later section. Sand in suspension is trans- 

 ported through the breaker zone and into deeper water by rip currents. It is 

 l^robable that a significant net seaward transport of sediment occurs by a 

 general mixing between the waters of the surf zone, where the concentration of 

 suspended sediment is high, and the offshore waters, where the concentration 

 is relatively low. In addition it is hkely that auto-suspension (Section 5) of fine 

 sediments in the turbulent waters of the surf zone results in an outward diffusion 

 of the finer grades of sediment. 



E. Rise in Water Level 



Waves transmit momentum in the direction of their travel, the momentum 

 being relative, according to wave theories, to stiU water elsewhere. Also wave 

 theories assume the momentum flux to continue on indefinitely. When this on- 

 ward flux is prevented, however, the flux of momentum against the shore 

 constitutes a shoreward force. Under steady-state conditions the shoreward 

 force should be just balanced by a rise in water level on the beach. The rise 

 should be proportional to the component of momentum flux normal to the 

 sliore : Cpq cos a, where pq is the momentum per unit crest length, C is the 

 velocity of the breaking wave, and a is the angle between the breaking wave 

 and the shore. However, in nature the grouping of high waves followed by low 

 waves causes a pulsation of the water level in the surf zone, called surf beat 

 (Munk, 1949a). Little attention has been paid to the degree of water-level rise 

 which results under steady state of pulsating conditions, or to its gradient 

 relative to water depth offshore. In a study of the second-order effects of the 

 interaction between waves and currents Longuet-Higgins and Stewart 

 (1962) have evaluated the additional flux of momentiun due to the waves 

 in terms of a "radiation stress". Assuming no loss of energy by friction or 

 reflection, thev show that the radiation stress associated with waves of steady 



